Typically, the Vice President breaks a tie in the Senate. Article I, section 3, clause 6 of the Constitution specifies that “[t]he Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided.” In effect, it means that there should never be such a thing as a divided Senate, unless the Vice President is absent or exercising the Office of President of the United States.

Yet, in Federalist No. 69, Alexander Hamilton explicitly contemplates that when the Senate exercises its Appointments Clause “advice and consent” power, there may well be a divided Senate andthat when the Senate is “equally divided,” the appointment fails:

The President is to nominate, and, WITH THE ADVICE AND CONSENT OF THE SENATE, to appoint ambassadors and other public ministers, judges of the Supreme Court, and in general all officers of the United States established by law, and whose appointments are not otherwise provided for by the Constitution. . . . There is evidently a great inferiority in the power of the President, in this particular, to that of the British king; nor is it equal to that of the governor of New York, if we are to interpret the meaning of the constitution of the State by the practice which has obtained under it. The power of appointment [in New York] is . . . lodged in a council, composed of the governor and four members of the Senate, chosen by the Assembly. The governor CLAIMS, and has frequently EXERCISED, the right of nomination, and is ENTITLED to a casting vote in the appointment. If he really has the right of nominating, his authority is in this respect equal to that of the President, and exceeds it in the article of the casting vote. In the national government, if the Senate should be divided, no appointment could be made; in the government of New York, if the council should be divided, the governor can turn the scale, and confirm his own nomination.

At face value, Hamilton seemed to believe (a) that there may be a divided Senate in the appointments context; and (b) that a tie in the Senate defeats a President’s nomination.

I think this illustrates the difference between original meaning originalism and at least some forms of original intent originalism. Under an original intent approach, there might be something to this argument. Hamilton seemed to believe, writing this part of The Federalist (and perhaps elsewhere), that an evenly divided Senate would block an appointment. As the post further describes, there do not appear to be clear expressions of intent at the Convention to allow the Vice President to break appointments ties.

Under an original meaning approach, Hamilton's commentary is interesting but ultimately unpersuasive. The text of Article I, section 3, clause 6 ("The Vice President of the United States shall be President of the Senate, but shall have no Vote, unless they be equally divided") has the clear implication that the Vice President shall have a vote when the Senators are equally divided. Nothing in the text of the appointments clause suggests that the appointments clause is exempt from this rule. Giving the Vice President a tie-breaking role in appointments is far from an absurd design (especially when the Vice President was not selected by the President). Hamilton did not explain, in Federalist 69 or elsewhere, why Article I, section 3, clause 6 would not apply to the appointments clause, nor (so far as I'm aware) did anyone else in the founding era.

Most likely, Hamilton simply forgot about the Vice President's tiebreaker in writing Federalist 69. Hamilton wrote a prodigious amount. Much of it is extremely insightful and very focused on the Constitution's text. I've relied on Hamilton in much of my scholarship. But he made mistakes. No single source is definitive or infallible, and (to an original meaning originalist) the ultimate touchstone is the Constitution's text, not what one person said about it at the time. If the text seems clear, and a person contradicting its apparent clear meaning does not offer an explanation, this is not likely to sway an original meaning originalist from the text's clear meaning, even if the contradicting person is Hamilton. As Chief Justice Marshall wrote (in McCulloch v. Maryland):

the opinions expressed by the authors of that work [The Federalist] have been justly supposed to be entitled to great respect in expounding the Constitution. No tribute can be paid to them which exceeds their merit; but in applying their opinions to the cases which may arise in the progress of our government, a right to judge of their correctness must be retained.